What Is the Distributive Property Law in Mathematics?

The distributive property law of numbers is a handy way of simplifying complex mathematical equations by breaking them down into smaller parts. It can be especially useful if you are struggling to understand algebra. 

Adding and Multiplying

Students usually begin learning the distributive property law when they start advanced multiplication. Take, for instance, multiplying 4 and 53. Calculating this example will require carrying the number 1 when you multiply, which can be tricky if you're being asked to solve the problem in your head.

There's an easier way of solving this problem. Begin by taking the larger number and rounding it down to the nearest figure that's divisible by 10. In this case, 53 becomes 50 with a difference of 3. Next, multiply both numbers by 4, then add the two totals together. Written out, the calculation looks like this:

53 x 4 = 212, or
(4 x 50) + (4 x 3) = 212, or
200 + 12 = 212

Simple Algebra

The distributive property also can be used to simplify algebraic equations by eliminating the parenthetical portion of the equation. Take for instance the equation a(b + c), which also can be written as (ab) + (ac) because the distributive property dictates that a, which is outside the parenthetical, must be multiplied by both b and c. In other words, you are distributing the multiplication of a between both b and c. For example:

2(3+6) = 18, or
(2 x 3) + (2 x 6) = 18, or
6 + 12 = 18

Don't be fooled by the addition. It's easy to misread the equation as (2 x 3) + 6 = 12. Remember, you are distributing the process of multiplying 2 evenly between 3 and 6.

Advanced Algebra

The distributive property law can also be used when multiplying or dividing polynomials, which are algebraic expressions that include real numbers and variables, and monomials, which are algebraic expressions consisting of one term.

You can multiply a polynomial by a monomial in three simple steps using the same concept of distributing the calculation:

1. Multiply the outside term by the first term in parenthesis.
2. Multiply the outside term by the second term in parenthesis.
3. Add the two sums.

Written out, it looks like this:

x(2x+10), or
(x * 2x) + (x * 10), or
2​x² + 10x

To divide a polynomial by a monomial, split it up into separate fractions then reduce. For example:

(4x³ + 6x² + 5x) / x, or
(4x³ / x) + (6x² / x) + (5x / x), or
4x² + 6x + 5

You also can use the distributive property law to find the product of binomials, as shown here:

(x + y)(x + 2y), or
(x + y)x + (x + y)(2y), or
x​²+xy +2xy 2y^2, or
x² + 3xy +2y²

More Practice

These algebra worksheets will help you understand how the distributive property law works. The first four do not involve exponents, which should make it easier for students to understand the basics of this important mathematical concept.